Frustum
A standard method from Arcs, sectors and mensuration: the steps and worked examples.
Steps
Volume of frustum = volume of the big cone − volume of the small cone cut off the top. The small cone is similar to the big one.
- Complete the cone. Find the height (or radius) of the small cone with similar triangles.
- Work out both cone volumes, V = ⅓πr²h.
- Subtract: frustum = big cone − small cone. Check with the volume ratio = (scale factor)³.
Worked examples
Example 1: Cut the cone in example 2 halfway up, parallel to the base. Find the volume of the frustum.
small cone : big cone, length ratio = 1 : 2
volume ratio = 1 : 8
frustum = big − small = 96π × (1 − 1/8) = 84π cm³For the whole topic, read Arcs, sectors and mensuration: key points, common mistakes and how the DSE tests it.